Question

Water is flowing through a channel (lying in a vertical plane) as shown in the figure. Three sections $A, B$ and $C$ are shown. Sections $B$ and $C$ have equal area of cross section. If $P_{A}, P_{B}$ and $P_{C}$ are the pressures at $A, B$ and $C$. respectively then

• A. $P_{A} > P_{B}=P_{C}$
• B. $P_{A} < P_{B} < P_{C}$
• C. $P_{A} < P_{B}=P_{C}$
• D. $P_{A} > P_{B} > P_{C}$

Answer 1

Answer: (B)

Solution by using Bernoulli's principle and equation of continuity
Comparing points $A$ and $B$
$A_{A} v_{A}=A_{B} v_{B}\{$ equation of continuity $\}$
$\because A_{A} < A_{B}$
$v_{A} > v_{B}$
$P_{A}+\frac{1}{2} \rho V_{A}^{2}+\rho g h=P_{B}+\frac{1}{2} \rho V_{B}^{2}+\rho g h$
$\{$Bernoulli's equation$\}$
$\because v_{A} > v_{B}$
$\Rightarrow \frac{1}{2} \rho V_{A}^{2} >\frac{1}{2} \rho V_{B}^{2}$
$\therefore P_{A} < P_{B}$ ...(1)
Now comparing $C$ and $B$
$A_{B}=A_{C} \Rightarrow v_{B}=v_{C}$ [equation of continuity]
$P_{B}+\frac{1}{2} \rho V^{2}+\rho g h_{B}$
$=P_{C}+\frac{1}{2} \rho V^{2}+\rho g h_{C}$
$\Rightarrow P_{B}+\rho g h_{B}=P_{C}+\rho g h_{C}$
$\because h_{B} > h_{C}$ then ...(2)
$P_{B} < P_{C}$
Using (1) and (2)
We can say, $P_{A} < P_{B} < P_{C}$